Topological order and entanglement dynamics in the measurement-only XZZX quantum code

TL;DR

This paper investigates the entanglement and topological order dynamics in a measurement-only quantum circuit based on the [[5,1,3]] code, revealing phase transitions, critical behavior, and transient scaling regimes.

Contribution

It introduces a measurement-induced phase diagram for a stabilizer code circuit, analyzing topological and symmetry-breaking order disruptions and identifying novel critical and transient scaling behaviors.

Findings

Measurement breaks topological order and induces phase transition.

Rich phase diagram with simultaneous or separate order breaking.

Identification of a transient scaling regime in purification dynamics.

Abstract

We examine the dynamics of a $(1+1)$-dimensional measurement-only circuit defined by the stabilizers of the [[5,1,3]] quantum error correcting code interrupted by single-qubit Pauli measurements. The code corrects arbitrary single-qubit errors and it stabilizes an area law entangled state with a $D_2 = \mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry protected topological (SPT) order, as well as a symmetry breaking (SB) order from a two-fold bulk degeneracy. The Pauli measurements break the topological order and induce a phase transition into a trivial area law phase. Allowing more than one type of Pauli measurement increases the measurement-induced frustration, and the SPT and SB order can be broken either simultaneously or separately at nonzero measurement rate. This yields a rich phase diagram and unanticipated critical behavior at the phase transitions. Although the correlation length exponent $\nu=\tfrac43$ and the dynamical critical exponent $z=1$ are consistent with bond percolation, the prefactor of the logarithmic entanglement growth may take non-integer multiples of the percolation value. Remarkably, we identify a robust transient scaling regime for the purification dynamics of $L$ qubits. It reveals a modified dynamical critical exponent $z^*\neq z$, which is observable up to times $t\sim L^{z^*}$ and is reminiscent of the relaxation of critical systems into a prethermal state.